On 3-manifolds with finite solvable fundamental group

Inventiones mathematicae -     

Some 3-manifolds with 2-generated fundamental group

We describe a family of graph manifolds that have 2-generated fundamental group but do not admit a Heegaard splitting of genus 2. They are closely related to the examples provided by Boileau and Zieschang if these are considered as (non-proper) graph manifolds.Received: 14 August 2002     

Collapsing irreducible 3-manifolds with nontrivial fundamental group

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locally controlled, then M is a graph manifold. This is the last step in Perelman’s proof of Thurston’s Geometrisation Conjecture.     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

The structure of 3-manifolds with two-generated fundamental group

We provide a structure theorem for 3-manifolds with 2-generated fundamental group and non-trivial JSJ-decomposition. We further give a number of applications.     

Some 3-manifolds and 3-orbifolds with large fundamental group

We provide two new proofs of a theorem of Cooper, Long and Reid which asserts that, apart from an explicit finite list of exceptional manifolds, any compact orientable irreducible 3-manifold with non-empty boundary has large fundamental group. The first proof is direct and topological; the second is group-theoretic. These techniques are then applied to prove a string of results about (possibly closed) 3-orbifolds, which culminate in the following theorem. If is a knot in a compact orientable 3-manifold such that the complement of admits a complete finite-volume hyperbolic structure, then the orbifold obtained by assigning a singularity of order along has large fundamental group for infinitely many positive integers . We also obtain information about this set of values of . When is the 3-sphere, this has implications for the cyclic branched covers over the knot. In this case, we may also weaken the hypothesis that the complement of is hyperbolic to the assumption that is non-trivial.

     

Positive scalar curvature for manifolds with elementary abelian fundamental group

The statement often called the Gromov-Lawson-Rosenberg Conjecture asserts that a manifold with finite fundamental group should admit a metric of positive scalar curvature except when the -valued index of some Dirac operator with coefficients in a flat bundle is non-zero. We prove spin and oriented non-spin versions of this statement for manifolds (of dimension ) with elementary abelian fundamental groups , except for ``toral' classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of . The proofs involve the detailed structure of , as computed by Johnson and Wilson.

     

The rank of group of cyclotomic units in abelian fields

A formula about the rank of group of cyclotomic units in abelian fields is established. From that formula, a series of equivalent conditions for independence of the system of cyclotomic units in abelian fields is stated and proved. For the particular case of cyclotomic fields, further properties of the rank are researched.     

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over . The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.

     

On the abelian fundamental group scheme of a family of varieties

Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) induced by the canonical morphism from X to its Albanese scheme Alb _X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS _X/S ^τ )^⋎ → π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber X _η of X can be extended to a finite and commutative pointed torsor over X.     

Rings of quasi-endomorphisms of strongly indecomposable torsion-free abelian groups of rank 4 with pseudosocles of rank 3

In this paper, the classification of rings of quasi-endomorphisms of strongly indecomposable Abelian torsion-free groups of rank 4 with pseudosocles of rank 3 is obtained.     

Compact hyperbolic 4-manifolds of small volume

We prove the existence of a compact non-orientable hyperbolic 4-manifold of volume and a compact orientable hyperbolic 4-manifold of volume , obtainable from torsion-free subgroups of small index in the Coxeter group . At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.

     

Quasiendomorphism algebras of some quasidecomposable rank 4 torsion-free abelian groups

We classify the quasiendomorphism algebras of rank 4 torsion-free abelian groups quasidecomposable as the direct sum of strongly indecomposable rank 2 groups.     

Surgery of closed manifolds with dihedral fundamental group

In the paper the obstruction groups to obtaining simple homotopy equivalence by surgery from normal degree 1 maps of closed manifolds with dihedral fundamental group are computed. The cases of trivial orientation for the dihedral group and nontrivial orientation for the order 2 cyclic subgroup are considered. New results concerning the Browder-Livesey groups and natural maps ofL-groups arising in index 2 inclusions of the cyclic group into the dihedral group are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 238–250, August, 1998.The work of the first-named author was partially supported by of the Grant of the President of the Russian Federation, grant No. 96-15-96841. The work of the second-named author was partially suported the Ministry of Science and Technology of the Republic of Sloveniya, grant No. J1-7039-0101-95.     

action on the cohomology of a rank two abelian group with arbitrary coefficient domain

A rank two abelian group is in a natural way an -module. This induces an action of on its group cohomology for any trivial coefficient domain . In the present note we determine this module, including the question of when the universal coefficient theorem sequence splits.